System, Method, and recording medium for mirroring matrices for batched Cholesky decomposition on a graphic processing unit

ABSTRACT

A batched Cholesky decomposition method, system, and non-transitory computer readable medium for a Graphics Processing Unit (GPU), include mirroring a second problem matrix of a second problem to a first problem matrix of a first problem as paired matrices and shifting the second problem matrix by N+1 and combining the first problem matrix and the mirrored second problem matrix into one matrix of (N+1)×N by merging the first problem matrix and the mirrored second problem matrix. The first problem matrix and the second problem matrix are symmetric and positive definite matrices.

CROSS-REFERENCE TO RELATED APPLICATIONS

The present application is a Continuation Application of U.S. patentapplication Ser. No. 15/915,148, filed on Mar. 8, 2018, which is aContinuation Application of U.S. patent application Ser. No. 15/199,772,filed on Jun. 30, 2016, now U.S. Pat. No. 9,984,041, issued on May 29,2018, the entire contents of which are hereby incorporated by reference.

BACKGROUND

The present invention relates generally to a batched Choleskydecomposition method on a graphics processing unit (GPU), and moreparticularly, but not by way of limitation, to a system, method, andrecording medium for combining two symmetric and positive definite (SPD)matrices into one rectangular (or square) matrix to accelerate batcheddense Cholesky decomposition on a GPU by solving both symmetric positivedefinite matrices (e.g., two problems) at the same.

Rapid evolution of GPUs in performance, architecture, andprogrammability provides general and scientific computational potentialfar beyond their primary purpose, graphics processing. Conventionally,Cholesky decomposition has been considered as an algorithm for solvingsymmetric and positive definite linear systems using the GPU.

Cholesky decomposition is conventionally complex because the processrequires three-routines (e.g., square rooting, normalizing, andsubtracting inner product or updating a submatrix), the memory accesspattern is sub-optimal, and there is a high thread divergence.

Conventional techniques have attempted to improve Cholesky decompositionby, for example, forward and backward substitution, which can be usedfor various purposes such as for equalization, filtering data, andreconstructing data. Such techniques consider a way to speed up Choleskydecomposition by proposing a Single-Instruction-Multiple-Data(SIMD)-like special functionality, which requires a new-type of hardwareor modification to an existing hardware, and does not consider a batchedproblem.

FIG. 2 exemplary shows a Cholesky decomposition of related art. Asshown, updating a global memory ‘B’ is a problem because global memoryis not as efficient for processing as a shared memory ‘A’ (i.e.,on-chip, etc.) because there are fewer valid elements as the steps gofrom “step” 0 to “step i”. Also, there is thread divergence because anupdate (or no update) is based on the step. Further, there is an issueof load balancing as in the thread nearest “X” would do nothing after“step i” is complete and waits until “step 0” completes. This leads tosynchronization strain (e.g., for every computation, threesynchronizations are needed).

SUMMARY

However, the inventors have realized a non-abstract technical solutionto the technical problem to improve a computer-technology (e.g., aperformance of a GPU) by accelerating a batch of Cholesky decompositionon a GPU by mirroring and combining two problems (e.g., two symmetricand positive definite (SPD) matrices) to analyze and allocate the readdata to a thread and to a problem.

In an exemplary embodiment, the present invention can provide a batchedCholesky decomposition method for a Graphics Processing Unit (GPU)including at least a first problem and a second problem, the methodincluding mirroring a second problem matrix of the second problem to afirst problem matrix of the first problem, combining the first problemmatrix and the mirrored second problem matrix into a single problemmatrix, and allocating data read to a thread and to the first problemand the second problem, respectively.

Further, in another exemplary embodiment, the present invention canprovide a non-transitory computer-readable recording medium recording abatched Cholesky decomposition program for a Graphics Processing Unit(GPU) including at least a first problem and a second problem, theprogram causing a computer to perform: mirroring a second problem matrixof the second problem to a first problem matrix of the first problem,combining the first problem matrix and the mirrored second problemmatrix into a single problem matrix, and allocating data read to athread and to the first problem and the second problem, respectively.

Even further, in another exemplary embodiment, the present invention canprovide a batched Cholesky decomposition system for at least a firstproblem and a second problem on a Graphics Processing Unit (GPU), saidsystem including a processor, and a memory, the memory storinginstructions to cause the processor to: mirror a second problem matrixof the second problem to a first problem matrix of the first problem,combine the first problem matrix and the mirrored second problem matrixinto a single problem matrix, and allocate data read to a thread and tothe first problem and the second problem, respectively.

There has thus been outlined, rather broadly, an embodiment of theinvention in order that the detailed description thereof herein may bebetter understood, and in order that the present contribution to the artmay be better appreciated. There are, of course, additional exemplaryembodiments of the invention that will be described below and which willform the subject matter of the claims appended hereto.

It is to be understood that the invention is not limited in itsapplication to the details of construction and to the arrangements ofthe components set forth in the following description or illustrated inthe drawings. The invention is capable of embodiments in addition tothose described and of being practiced and carried out in various ways.In addition, it is to be understood that the phraseology and terminologyemployed herein, as well as the abstract, are for the purpose ofdescription and should not be regarded as limiting.

As such, those skilled in the art will appreciate that the conceptionupon which this disclosure is based may readily be utilized as a basisfor the designing of other structures, methods and systems for carryingout the several purposes of the present invention. It is important,therefore, that the claims be regarded as including such equivalentconstructions insofar as they do not depart from the spirit and scope ofthe present invention.

BRIEF DESCRIPTION OF THE DRAWINGS

The exemplary aspects of the invention will be better understood fromthe following detailed description of the exemplary embodiments of theinvention with reference to the drawings.

FIG. 1 exemplarily shows a high-level flow chart for a batched Choleskydecomposition method 100.

FIG. 2 exemplarily shows a Cholesky decomposition method of a RelatedArt.

FIG. 3 exemplarily shows an exemplary working embodiment of Steps 101and 102.

FIG. 4 exemplarily shows an exemplary working embodiment of Step 103 anda working embodiment of the batched Cholesky decomposition method 100.

FIG. 5 depicts a cloud-computing node according to an embodiment of thepresent invention.

FIG. 6 depicts a cloud-computing environment according to anotherembodiment of the present invention.

FIG. 7 depicts abstraction model layers according to an embodiment ofthe present invention.

DETAILED DESCRIPTION

The invention will now be described with reference to FIGS. 1-7, inwhich like reference numerals refer to like parts throughout. It isemphasized that, according to common practice, the various features ofthe drawing are not necessarily to scale. On the contrary, thedimensions of the various features can be arbitrarily expanded orreduced for clarity. Exemplary embodiments are provided below forillustration purposes and do not limit the claims.

With reference now to FIG. 1, the Batched Cholesky decomposition method100 includes various steps to pair matrices to accelerate batched denseCholesky decomposition on a GPU. As shown in at least FIG. 6, one ormore computers of a computer system 12 can include a memory 28 havinginstructions stored in a storage system to perform the steps of FIG. 1.

Although as shown in FIG(S). 5-7 and as described later, the computersystem/server 12 is exemplarily shown in cloud computing node 10 as ageneral-purpose computing circuit which may execute in a layer thebatched Cholesky decomposition system method (FIG. 6), it is noted thatthe present invention can be implemented outside of the cloudenvironment.

The method 100 is used for Cholesky decomposition to solve a linearsystem on a GPU to solve a batch of problems. The linear systemcomprises “Ax=B” where A is symmetric and positive definite (e.g.,LL{circumflex over ( )}x=B). That is, referring to FIG. 3, the batch ofproblems includes at least a first problem “Problem 1” and a secondproblem “Problem 2”.

That is, the steps of the method 100 are all part of an improved methodfor batched Cholesky decomposition.

As shown in FIG. 3, Step 101 mirrors (folds) the second problem to thefirst problem and shifts the second problem by ‘N+1’. That is, becausethe matrices of Cholesky decomposition are symmetrical (e.g., A=LL*),the second problem can be mirrored by Step 101 to provide an acceptablematrix for the Cholesky decomposition. It is noted that “mirroring” or“folding” can be used interchangeably and have the same meaning.

Step 102 combines the first problem and the mirror second problem into asingle matrix. The combined matrix comprising the first problem and themirrored second problem has a memory layout of an (N+1)×N matrix. Step102 shifts the mirrored second problem by (N+1) such that the memorylayout comprises the (N+1)×N matrix (e.g., a rectangular or squarematrix). That is, Step 102 combines (e.g., pairs) the first problem andthe mirrored second problem into one square matrix when at least twoproblems are present to be solved.

As shown in FIG. 4, the combined matrix comprises global memory ‘B’ andshared memory ‘A’ for the first problem which has regular intervals andshared memory ‘C’ for the second problem which is continuous. BecauseStep 102 combines the first problem and the mirrored second problem intoa square matrix of (N+1)×N, each thread performs a same amount of workin each step (e.g., each step has a same amount of elements).

Step 103 allocates the data read from the global memory to a thread andto the respective problem (e.g., a thread can include read data from thefirst problem and the second problem). Once the read data is allocatedto the thread, the thread will update the values in the data byperforming the Cholesky decomposition algorithm. In other words, thedata read from global memory is needed for sub-matrix update which isthe last step for a given “X” or “Y” diagonal element.

Thereby, mirroring (folding) the second problem matrix and combining themirrored second problem matrix with the first problem matrix as above,Step 103 can read the fixed size data length with a fixed data intervalfor both problems, instead of reading twice for each problem in anirregular fashion.

That is, each step includes elements of global memory from the firstproblem and the second problem because each step is a same size as aresult of the combined problems. Step 103 reads all data in the stepsand allocates which problem that the read data belongs (e.g., eachthread has an equal amount of work to do during the Choleskydecomposition). When the Cholesky decomposition algorithm is performedon the read data, the threads update the values of the steps. Also, Step103 does not discard data and does not have to wait for steps to be readbecause all data is read at a same time. Thus, there is no idle time.

Further, the method 100 requires only a single synchronization for everytwo problems when the Cholesky decomposition algorithm is performed(e.g., 50% reduction in synchronization and divergence when compared toconventional techniques). In addition, because each step has a samenumber of elements, each thread has a same number of processes toperform in Step 103 when allocating the data read from the global memorythereby increasing a global memory efficiency. That is, there are noidle time of threads when the Cholesky decomposition algorithm isperformed on the global memory data.

For example, on a k40m GPU with 20,000 problems, the method 100 has a3.3-times increase in speed over Net_64 (e.g., 0.73 to 0.22) and a10-times increase in speed over Yahoo_64 (e.g., 0.95 to 0.095).

It is noted that the method 100 can be implemented on a standard(“off-the-shelf”) GPU without any hardware modifications to the GPU.

Exemplary Hardware Aspects, Using a Cloud Computing Environment

It is understood in advance that although this disclosure includes adetailed description on cloud computing, implementation of the teachingsrecited herein are not limited to a cloud-computing environment. Rather,embodiments of the present invention are capable of being implemented inconjunction with any other type of computing environment now known orlater developed.

Cloud computing is a model of service delivery for enabling convenient,on-demand network access to a shared pool of configurable computingresources (e.g. networks, network bandwidth, servers, processing,memory, storage, applications, virtual machines, and services) that canbe rapidly provisioned and released with minimal management effort orinteraction with a provider of the service. This cloud model may includeat least five characteristics, at least three service models, and atleast four deployment models.

Characteristics are as follows:

On-demand self-service: a cloud consumer can unilaterally provisioncomputing capabilities, such as server time and network storage, asneeded automatically without requiring human interaction with theservice's provider.

Broad network access: capabilities are available over a network andaccessed through standard mechanisms that promote use by heterogeneousthin or thick client platforms (e.g., mobile phones, laptops, and PDAs).

Resource pooling: the provider's computing resources are pooled to servemultiple consumers using a multi-tenant model, with different physicaland virtual resources dynamically assigned and reassigned according todemand. There is a sense of location independence in that the consumergenerally has no control or knowledge over the exact location of theprovided resources but may be able to specify location at a higher levelof abstraction (e.g., country, state, or datacenter).

Rapid elasticity: capabilities can be rapidly and elasticallyprovisioned, in some cases automatically, to quickly scale out andrapidly released to quickly scale in. To the consumer, the capabilitiesavailable for provisioning often appear to be unlimited and can bepurchased in any quantity at any time.

Measured service: cloud systems automatically control and optimizeresource use by leveraging a metering capability at some level ofabstraction appropriate to the type of service (e.g., storage,processing, bandwidth, and active user accounts). Resource usage can bemonitored, controlled, and reported providing transparency for both theprovider and consumer of the utilized service.

Service Models are as follows:

Software as a Service (SaaS): the capability provided to the consumer isto use the provider's applications running on a cloud infrastructure.The applications are accessible from various client circuits through athin client interface such as a web browser (e.g., web-based e-mail).The consumer does not manage or control the underlying cloudinfrastructure including network, servers, operating systems, storage,or even individual application capabilities, with the possible exceptionof limited user-specific application configuration settings.

Platform as a Service (PaaS): the capability provided to the consumer isto deploy onto the cloud infrastructure consumer-created or acquiredapplications created using programming languages and tools supported bythe provider. The consumer does not manage or control the underlyingcloud infrastructure including networks, servers, operating systems, orstorage, but has control over the deployed applications and possiblyapplication hosting environment configurations.

Infrastructure as a Service (IaaS): the capability provided to theconsumer is to provision processing, storage, networks, and otherfundamental computing resources where the consumer is able to deploy andrun arbitrary software, which can include operating systems andapplications. The consumer does not manage or control the underlyingcloud infrastructure but has control over operating systems, storage,deployed applications, and possibly limited control of select networkingcomponents (e.g., host firewalls).

Deployment Models are as follows:

Private cloud: the cloud infrastructure is operated solely for anorganization. It may be managed by the organization or a third party andmay exist on-premises or off-premises.

Community cloud: the cloud infrastructure is shared by severalorganizations and supports a specific community that has shared concerns(e.g., mission, security requirements, policy, and complianceconsiderations). It may be managed by the organizations or a third partyand may exist on-premises or off-premises.

Public cloud: the cloud infrastructure is made available to the generalpublic or a large industry group and is owned by an organization sellingcloud services.

Hybrid cloud: the cloud infrastructure is a composition of two or moreclouds (private, community, or public) that remain unique entities butare bound together by standardized or proprietary technology thatenables data and application portability (e.g., cloud bursting forload-balancing between clouds).

A cloud computing environment is service oriented with a focus onstatelessness, low coupling, modularity, and semantic interoperability.At the heart of cloud computing is an infrastructure comprising anetwork of interconnected nodes.

Referring now to FIG. 5, a schematic of an example of a cloud computingnode is shown. Cloud computing node 10 is only one example of a suitablecloud computing node and is not intended to suggest any limitation as tothe scope of use or functionality of embodiments of the inventiondescribed herein. Regardless, cloud computing node 10 is capable ofbeing implemented and/or performing any of the functionality set forthhereinabove.

In cloud computing node 10 there is a computer system/server 12, whichis operational with numerous other general purpose or special purposecomputing system environments or configurations. Examples of well-knowncomputing systems, environments, and/or configurations that may besuitable for use with computer system/server 12 include, but are notlimited to, personal computer systems, server computer systems, thinclients, thick clients, hand-held or laptop circuits, multiprocessorsystems, microprocessor-based systems, set top boxes, programmableconsumer electronics, network PCs, minicomputer systems, mainframecomputer systems, and distributed cloud computing environments thatinclude any of the above systems or circuits, and the like.

Computer system/server 12 may be described in the general context ofcomputer system-executable instructions, such as program modules, beingexecuted by a computer system. Generally, program modules may includeroutines, programs, objects, components, logic, data structures, and soon that perform particular tasks or implement particular abstract datatypes. Computer system/server 12 may be practiced in distributed cloudcomputing environments where tasks are performed by remote processingcircuits that are linked through a communications network. In adistributed cloud computing environment, program modules may be locatedin both local and remote computer system storage media including memorystorage circuits.

As shown in FIG. 5, computer system/server 12 in cloud computing node 10is shown in the form of a general-purpose computing circuit. Thecomponents of computer system/server 12 may include, but are not limitedto, one or more processors or processing units 16, a system memory 28,and a bus 18 that couples various system components including systemmemory 28 to processor 16.

Bus 18 represents one or more of any of several types of bus structures,including a memory bus or memory controller, a peripheral bus, anaccelerated graphics port, and a processor or local bus using any of avariety of bus architectures. By way of example, and not limitation,such architectures include Industry Standard Architecture (ISA) bus,Micro Channel Architecture (MCA) bus, Enhanced ISA (EISA) bus, VideoElectronics Standards Association (VESA) local bus, and PeripheralComponent Interconnects (PCI) bus.

Computer system/server 12 typically includes a variety of computersystem readable media. Such media may be any available media that isaccessible by computer system/server 12, and it includes both volatileand non-volatile media, removable and non-removable media.

System memory 28 can include computer system readable media in the formof volatile memory, such as random access memory (RAM) 30 and/or cachememory 32. Computer system/server 12 may further include otherremovable/non-removable, volatile/non-volatile computer system storagemedia. By way of example only, storage system 34 can be provided forreading from and writing to a non-removable, non-volatile magnetic media(not shown and typically called a “hard drive”). Although not shown, amagnetic disk drive for reading from and writing to a removable,non-volatile magnetic disk (e.g., a “floppy disk”), and an optical diskdrive for reading from or writing to a removable, non-volatile opticaldisk such as a CD-ROM, DVD-ROM or other optical media can be provided.In such instances, each can be connected to bus 18 by one or more datamedia interfaces. As will be further depicted and described below,memory 28 may include at least one program product having a set (e.g.,at least one) of program modules that are configured to carry out thefunctions of embodiments of the invention.

Program/utility 40, having a set (at least one) of program modules 42,may be stored in memory 28 by way of example, and not limitation, aswell as an operating system, one or more application programs, otherprogram modules, and program data. Each of the operating system, one ormore application programs, other program modules, and program data orsome combination thereof, may include an implementation of a networkingenvironment. Program modules 42 generally carry out the functions and/ormethodologies of embodiments of the invention as described herein.

Computer system/server 12 may also communicate with one or more externalcircuits 14 such as a keyboard, a pointing circuit, a display 24, etc.;one or more circuits that enable a user to interact with computersystem/server 12; and/or any circuits (e.g., network card, modem, etc.)that enable computer system/server 12 to communicate with one or moreother computing circuits. Such communication can occur via Input/Output(I/O) interfaces 22. Still yet, computer system/server 12 cancommunicate with one or more networks such as a local area network(LAN), a general wide area network (WAN), and/or a public network (e.g.,the Internet) via network adapter 20. As depicted, network adapter 20communicates with the other components of computer system/server 12 viabus 18. It should be understood that although not shown, other hardwareand/or software components could be used in conjunction with computersystem/server 12. Examples, include, but are not limited to: microcode,circuit drivers, redundant processing units, external disk drive arrays,RAID systems, tape drives, and data archival storage systems, etc.

Referring now to FIG. 6, illustrative cloud computing environment 50 isdepicted. As shown, cloud computing environment 50 comprises one or morecloud computing nodes 10 with which local computing circuits used bycloud consumers, such as, for example, personal digital assistant (PDA)or cellular telephone 54A desktop computer 54B, laptop computer 54C,and/or automobile computer system 54N may communicate. Nodes 10 maycommunicate with one another. They may be grouped (not shown) physicallyor virtually, in one or more networks, such as Private, Community,Public, or Hybrid clouds as described hereinabove, or a combinationthereof. This allows cloud computing environment 50 to offerinfrastructure, platforms and/or software as services for which a cloudconsumer does not need to maintain resources on a local computingcircuit. It is understood that the types of computing circuits 54A-Nshown in FIG. 6 are intended to be illustrative only and that computingnodes 10 and cloud computing environment 50 can communicate with anytype of computerized circuit over any type of network and/or networkaddressable connection (e.g., using a web browser).

Referring now to FIG. 7, a set of functional abstraction layers providedby cloud computing environment 50 (FIG. 6) is shown. It should beunderstood in advance that the components, layers, and functions shownin FIG. 7 are intended to be illustrative only and embodiments of theinvention are not limited thereto. As depicted, the following layers andcorresponding functions are provided:

Hardware and software layer 60 includes hardware and softwarecomponents. Examples of hardware components include: mainframes 61; RISC(Reduced Instruction Set Computer) architecture based servers 62;servers 63; blade servers 64; storage circuits 65; and networks andnetworking components 66. In some embodiments, software componentsinclude network application server software 67 and database software 68.

Virtualization layer 70 provides an abstraction layer from which thefollowing examples of virtual entities may be provided: virtual servers71; virtual storage 72; virtual networks 73, including virtual privatenetworks; virtual applications and operating systems 74; and virtualclients 75.

In one example, management layer 80 may provide the functions describedbelow. Resource provisioning 81 provides dynamic procurement ofcomputing resources and other resources that are utilized to performtasks within the cloud computing environment. Metering and Pricing 82provide cost tracking as resources are utilized within the cloudcomputing environment, and billing or invoicing for consumption of theseresources. In one example, these resources may comprise applicationsoftware licenses. Security provides identity verification for cloudconsumers and tasks, as well as protection for data and other resources.User portal 83 provides access to the cloud computing environment forconsumers and system administrators. Service level management 84provides cloud computing resource allocation and management such thatrequired service levels are met. Service Level Agreement (SLA) planningand fulfillment 85 provide pre-arrangement for, and procurement of,cloud computing resources for which a future requirement is anticipatedin accordance with an SLA.

Workloads layer 90 provides examples of functionality for which thecloud computing environment may be utilized. Examples of workloads andfunctions which may be provided from this layer include: mapping andnavigation 91; software development and lifecycle management 92; virtualclassroom education delivery 93; data analytics processing 94;transaction processing 95; and, more particularly relative to thepresent invention, the anti-counterfeiting system 100 and theanti-counterfeiting system 600 described herein.

The descriptions of the various embodiments of the present inventionhave been presented for purposes of illustration, but are not intendedto be exhaustive or limited to the embodiments disclosed. Manymodifications and variations will be apparent to those of ordinary skillin the art without departing from the scope and spirit of the describedembodiments. The terminology used herein was chosen to best explain theprinciples of the embodiments, the practical application or technicalimprovement over technologies found in the marketplace, or to enableothers of ordinary skill in the art to understand the embodimentsdisclosed herein.

Further, Applicant's intent is to encompass the equivalents of all claimelements, and no amendment to any claim of the present applicationshould be construed as a disclaimer of any interest in or right to anequivalent of any element or feature of the amended claim.

What is claimed is:
 1. A batched Cholesky decomposition method for aGraphics Processing Unit (GPU), the method comprising: mirroring asecond problem matrix of a second problem to a first problem matrix of afirst problem as paired matrices and shifting the second problem matrixby N+1; and combining the first problem matrix and the mirrored secondproblem matrix into one matrix of (N+1)×N by merging the first problemmatrix and the mirrored second problem matrix, wherein the first problemmatrix and the second problem matrix are symmetric and positive definitematrices, and wherein the GPU performs batched dense Choleskydecomposition with the one matrix from the combining to accelerate theCholesky decomposition.
 2. A non-transitory computer-readable recordingmedium recording a batched Cholesky decomposition program for a GraphicsProcessing Unit (GPU), the program causing a computer to perform:mirroring a second problem matrix of a second problem to a first problemmatrix of a first problem as paired matrices and shifting the secondproblem matrix by N+1; and combining the first problem matrix and themirrored second problem matrix into one matrix of (N+1)×N by merging thefirst problem matrix and the mirrored second problem matrix, wherein thefirst problem matrix and the second problem matrix arc symmetric andpositive definite matrices, and wherein the GPU performs batched denseCholesky decomposition with the one matrix from the combining toaccelerate the Cholesky decomposition.
 3. A batched Choleskydecomposition system on a Graphics Processing Unit (GPU), said systemcomprising: a processor; and a memory, the memory storing instructionsto cause the processor to: mirroring a second problem matrix of a secondproblem to a first problem matrix of a first problem as paired matricesand shifting the second problem matrix by N+1; and combining the firstproblem matrix and the mirrored second problem matrix into one matrix of(N+1)×N by merging the first problem matrix and the mirrored secondproblem matrix, wherein the first problem matrix and the second problemmatrix are symmetric and positive definite matrices, and wherein the GPUperforms batched dense Cholesky decomposition with the one matrix fromthe combining to accelerate Cholesky decomposition.